Could local zeta functions take the role of Riemann Zeta in TGD framework?

The recent view about TGD (for a summary see this, this, and this) leads to some conjectures about Riemann Zeta.

1. Non-trivial zeros should be algebraic numbers.

2. The building blocks in the product decomposition of ζ should be algebraic numbers for non-trivial zeros of zeta.

3. The values of zeta for their combinations with positive imaginary part with positive integer coefficients should be algebraic numbers.

These conjectures are motivated by the findings that Riemann Zeta seems to be associated with critical systems and by the fact that non-trivial zeros of zeta are analogous to complex conformal weights. The necessity to make such a strong conjectures, in particular the third conjecture, is an unsatisfactory feature of the theory and one could ask how to modify this picture. Also a clear physical interpretation of Riemann zeta is lacking.

1. Local zeta functions and Weil conjectures

Riemann Zeta is not the only zeta (see this and this). There is entire zoo of zeta functions and the natural question is whether some other zeta sharing the basic properties of Riemann zeta having zeros at critical line could be more appropriate in TGD framework.

The so called local zeta functions analogous to the factors ζ_p(s)= 1/(1-p^-s) of Riemann Zeta can be used to code algebraic data about say numbers about solutions of algebraic equations reduced to finite fields. The local zeta functions appearing in Weil's conjectures associated with finite fields G(p,k) and thus to single prime. The extensions G(p,nk) of this finite field are considered. These local zeta functions code the number for the points of algebraic variety for given value of n. Weil's conjectures also state that if X is a mod p reduction of non-singular complex projective variety then the degree for the polynomial multiplying the product ζ(s)×ζ(s-1) equals to Betti number. Betti number is 2 times genus in 2-D case.

It has been proven that the zetas of Weil are associated with single prime p, they satisfy functional equation, their zeros are at critical lines, and rather remarkably, they are rational functions of p^-s. For instance, for elliptic curves zeros are at critical line.

The general form for the local zeta is ζ(s)= exp(G(s)), where G= ∑ g_np^-ns, g_n=N_n/n, codes for the numbers N_n of points of algebraic variety for n^th extension of finite field F with nk elements assuming that F has k=p^r elements. This transformation resembles the relationship Z= exp(F) between partition function and free energy Z= exp(F) in thermodynamics.

The exponential form is motivated by the possibility to factorize the zeta function into a product of zeta functions. Note also that in the situation when N_n approaches constant N_∞, the division of N_n by n gives essentially 1/(1-N_∞p^-s) and one obtains the factor of Riemann Zeta at a shifted argument s-log^p(N_∞). The local zeta associated with Riemann Zeta corresponds to N_n=1.

2. Local zeta functions and TGD

The local zetas are associated with single prime p, they satisfy functional equation, their zeros lie at the critical lines, and they are rational functions of p^-s. These features are highly desirable from the TGD point of view.

2.1 Why local zeta functions are natural in TGD framework?

In TGD framework modified Dirac equation assigns to a partonic 2-surface a p-adic prime p and inverse of the zeta defines local conformal weight (see this). The intersection of the real and corresponding p-adic parton 2-surface is the set containing the points that one is interested in. Hence local zeta sharing the basic properties of Riemann zeta is highly desirable and natural. In particular, if the local zeta is a rational function then the inverse images of rational points of the geodesic sphere are algebraic numbers. Of course, one might consider a stronger constraint that the inverse image is rational. Note that one must still require that p^-s as well as s are algebraic numbers for the zeros of the local zeta (the first two conditions listed in the beginning) if one wants the number theoretical universality.

Since the modified Dirac operator assigns to a given partonic 2-surface a p-adic prime p (see this), one can ask whether the inverse ζ_p^-1(z) of some kind of local zeta directly coding data about partonic 2-surface could define the generalized eigenvalues of the modified Dirac operator and radial super-canonical conformal weights so that the conjectures about Riemann Zeta would not be needed at all.

The eigenvalues of the modified Dirac operator (see this) would in a holographic manner code for information about partonic 2-surface. This kind of algebraic geometric data are absolutely relevant for TGD since U-matrix and probably also S-matrix must be formulated in terms of the data related to the intersection of real and partonic 2-surfaces (number theoretic braids) obeying same algebraic equations and consisting of algebraic points in the appropriate algebraic extension of p-adic numbers. Note that the hierarchy of algebraic extensions of p-adic number fields would give rise to a hierarchy of zetas so that the algebraic extension used would directly reflect itself in the eigenvalue spectrum of the modified Dirac operator and super-canonical conformal weights. This is highly desirable but not achieved if one uses Riemann Zeta.

One must of course leave open the possibility that for real-real transitions the inverse of the zeta defined as a product of the local zetas (very much analogous to Riemann Zeta) defines the conformal weights. This kind of picture would conform with the idea about real physics as a kind of adele formed from p-adic physics.

2.2 Finite field hierarchy is not natural in TGD context

That local zeta functions are assigned with a hierarchy of finite field extensions do not look natural in TGD context. The reason is that these extensions are regarded as abstract extensions of G(p,k) as opposed to a large number of algebraic extensions isomorphic with finite fields as abstract number fields and induced from the extensions of p-adic number fields. Sub-field property is clearly highly relevant in TGD framework just as the sub-manifold property is crucial for geometrizing also other interactions than gravitation in TGD framework.

The O(pⁿ) hierarchy for the p-adic cutoffs would naturally replace the hierarchy of finite fields. This hierarchy is quite different from the hierarchy of finite fields since one expects that the number of solutions becomes constant at the limit of large n and also at the limit of large p so that powers in the function G coding for the numbers of solutions of algebraic equations as function of n should not increase but approach constant N_∞. The possibility to factorize exp(G) to a product exp(G₀)exp(G_∞) would mean a reduction to a product of a rational function and factor(s) ζ_p(s)=1/(1-p^{-s₁}) associated with Riemann Zeta with argument s shifted to s₁=s-log_p(N_∞).

2.3 What data local zetas could code?

The next question is what data the local zeta functions could code.

1. It is not at clear whether it is useful to code global data such as the numbers of points of partonic 2-surface modulo pⁿ. The notion of number theoretic braid occurring in the proposed approach to S-matrix (see this) suggests that the zeta at an algebraic point z of the geodesic sphere S² of CP₂ or of light-cone boundary should code purely local data such as the numbers N_n of points which project to z as function of p-adic cutoff pⁿ. In the generic case this number would be finite for non-vacuum extremals with 2-D S² projection. The n^th coefficient g_n=N_n/n of the function G_p would code the number N_n of these points in the approximation O(pⁿ⁺¹)=0 for the algebraic equations defining the p-adic counterpart of the partonic 2-surface.

2. In a region of partonic 2-surface where the numbers N_n of these points remain constant, ζ(s) would have constant functional form and therefore the information in this discrete set of algebraic points would allow to deduce deduce information about the numbers N_n. Both the algebraic points and generalized eigenvalues would carry the algebraic information.

3. A rather fascinating self referentiality would result: the generalized eigen values of the modified Dirac operator expressible in terms of inverse of zeta would code data for a sequence of approximations for the p-adic variant of the partonic 2-surface. This would be natural since second quantized induced spinor fields are correlates for logical thought in TGD inspired theory of consciousness. Even more, the data would be given at points ζ(s), s a rational value of a super-canonical conformal weight or a value of generalized eigenvalue of modified Dirac operator (which is essentially function s= ζ_p^-1(z) at geodesic sphere of CP₂ or of light-cone boundary).

For more details see the end of the chapter Construction of Quantum Theory: Symmetries of "Towards S-Matrix" and the article Topological Geometrodynamics: an Overall View. See also the article Could local zeta functions take the role of Riemann Zeta in TGD framework?.